Sorrento to Rome

Take the Circumvesuviana or Campania Express trains from Sorrento to Naples Garibaldi.
Transfer upstairs to Naples Centrale. Take the train from Naples Centrale to Rome.

edit: To clarify it Takes about 3 hours. from Sorrento to Naples using the above method

To add to Jazz' post, the Campania Express is a 2nd option to the Circumvesuviana. It travels the SAME route between Sorrento and Naples but with fewer stops. It's also a nicer train, operates seasonally (mid-March - October), and is a bit more expensive (still cheap) but only makes 4 or so runs a day so not as convenient. Info here:

https://www.sorrentoinsider.com/en/campania-express-train-schedule?timetable_route=sorrento_napoli

This train could also be used for your journey from Naples to Sorrento, or for day-tripping to Pompei, if timing works.

Just did this two weeks ago. Easiest way not cheapest. Take a private car from Sorrento to the train station in Naples. I believe the fast train was about an hour and a half not three hours.

SA, the question was to Rome. Add together the train from Sorrento, the train from Naples and the waiting for it and you are the thick edge of 3 plus hours.

Any other way will be slower unless you hire a helicopter. While the train is slow on the Sorrento peninsula so is the road, then from Naples you are looking at 300 km/h to Rome on the train. No car can get close in speed.